Ordered Size Ramsey Number of Paths

TL;DR

This paper investigates the ordered size Ramsey number for paths, establishing bounds that relate the number to the path lengths and extending recent results in oriented graph Ramsey theory.

Contribution

It provides new bounds on the ordered size Ramsey number of paths, connecting it to path lengths and logarithmic factors, and extends recent oriented graph results.

Findings

Lower bound:  r^2 s

Upper bound: C r^2 s ( log s)^3

Extends results from oriented graph Ramsey theory

Abstract

An ordered graph is a simple graph with an ordering on its vertices. Define the ordered path $P_n$ to be the monotone increasing path with $n$ edges. The ordered size Ramsey number $\tilde{r}(P_r,P_s)$ is the minimum number $m$ for which there exists an ordered graph $H$ with $m$ edges such that every two-coloring of the edges of $H$ contains a red copy of $P_r$ or a blue copy of $P_s$. For $2\leq r\leq s$, we show $\frac{1}{8}r^2s\leq \tilde{r}(P_r,P_s)\leq Cr^2s(\log s)^3$, where $C>0$ is an absolute constant. This problem is motivated by the recent results of Buci\'c-Letzter-Sudakov and Letzter-Sudakov for oriented graphs.